The notion of shape fibration with the near lifting of near multivalued paths property is studied. The relation of these maps—which agree with shape fibrations having totally disconnected fibers—with Hurewicz fibrations with the unique path lifting property is completely settled. Some results concerning homotopy and shape groups are presented for shape fibrations with the near lifting of near multivalued paths property. It is shown that for this class of shape fibrations the existence of liftings of a fine multivalued map is equivalent to an algebraic problem relative to the homotopy, shape or strong shape groups associated.
